×

zbMATH — the first resource for mathematics

Unsupervised curve clustering using B-splines. (English) Zbl 1039.91067
The model of observations is \(y_j^i=G^i(x_j^i)+\varepsilon_J^i\), where \(G^i\), \(i=1,\dots,n\) is the \(i\)-th experimental curve, \((y_j^i,x_j^i)\) are observations, \(\varepsilon_j^i\) are i.i.d. errors. The authors propose to fit \(G^i\) by B-splines and then to partition the coefficient vectors \(\beta^i\) of these splines by \(k\)-means clustering. It is shown that under suitable assumptions this procedure is consistent, i.e. the sequence of the cluster centres (means) tends to the minimizer of the sum of \(L_2\) distances between \(G^i\) and the nearest centre. This technique is applied to a data on cheese production.

MSC:
91C20 Clustering in the social and behavioral sciences
62G08 Nonparametric regression and quantile regression
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bock H. H., Advances in data science and classification pp 265– (1998) · doi:10.1007/978-3-642-72253-0_37
[2] Curry H. B., J. Anal. Math. 17 pp 71– (1966)
[3] De Boor C., A practical guide to splines (1978) · Zbl 0406.41003 · doi:10.1007/978-1-4612-6333-3
[4] Diday E., Elements d’analyse de donnees (1983)
[5] Hartigan J. A., Clustering algorithms (1975) · Zbl 0372.62040
[6] Hartigan J. A., J. Appl. Statist. 28 pp 100– (1979)
[7] Lemaire J., Statistiques et analyse des donnees 8 pp 41– (1983)
[8] Muller C., journees de statistique pp 600– (1996)
[9] Pollard D., Ann. Statist. 9 pp 135– (1981)
[10] Pollard D., Convergence of stochastic processes (1984) · Zbl 0544.60045 · doi:10.1007/978-1-4612-5254-2
[11] Ramsay J., Functional data analysis (1997) · Zbl 0882.62002 · doi:10.1007/978-1-4757-7107-7
[12] Ripley B. D., Pattern recognition and neural networks (1996) · Zbl 0853.62046 · doi:10.1017/CBO9780511812651
[13] Rockafellar R. T., Variational analysis (1998) · Zbl 0888.49001 · doi:10.1007/978-3-642-02431-3
[14] Schumaker L. L., Spline functions: basic theory (1981) · Zbl 0449.41004
[15] Geer S., Ann. Statist. 15 pp 587– (1987)
[16] Geer S., Empirical processes in M-estimation (2000)
[17] Vaart A. W., Weak convergence and empirical processes with applications to statistics (1996) · Zbl 0862.60002 · doi:10.1007/978-1-4757-2545-2
[18] Wegman E. J., J. Amer. Statist. Assoc. 78 pp 351– (1983)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.